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在宇宙学框架中,Noether对称技术揭示了一种有用的工具,可以检查精确的解。 在这项工作中,我们首先介绍约旦框架拉格朗日并应用共形变换以获得与爱因斯坦框架形式等效的拉格朗日。 然后,我们通过集中于爱因斯坦框架形式的单场场景,使用Noether对称方法分析宇宙α吸引子中场的动力学。 我们表明,利用Noether对称性,可以完全集成相应的动力学系统,并且可以精确获得对称性所显示的电势。 通过正确选择参数,比例因子的行为将在当前时期显示指数(de Sitter)行为。 此外,我们发现哈勃参数强烈取决于Noether对称性所显示的参数的初始值。 有趣的是,在所有情况下,它都可以延迟发展并在当前时代变成一个常数。
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Available online at www.sciencedirect.com
ScienceDirect
Nuclear Physics B 931 (2018) 216–225
www.elsevier.com/locate/nuclphysb
Noether symmetry approach in the cosmological
alpha-attractors
Narakorn Kaewkhao
a
, Thanyagamon Kanesom
a
,
Phongpichit Channuie
b,∗
a
Department of Physics, Faculty of Science, Prince of Songkla University, Hatyai 90112, Thailand
b
School of Science, Walailak University, Thasala, Nakhon Si Thammarat, 80160, Thailand
Received 19
January 2018; received in revised form 10 April 2018; accepted 16 April 2018
Available
online 21 April 2018
Editor: Stephan
Stieberger
Abstract
In
cosmological framework, Noether symmetry technique has revealed a useful tool in order to exam-
ine
exact solutions. In this work, we first introduce the Jordan-frame Lagrangian and apply the conformal
transformation in order to obtain the Lagrangian equivalent to Einstein-frame form. We then analyze the
dynamics of the field in the cosmological alpha-attractors using the Noether symmetry approach by focus-
ing
on the single field scenario in the Einstein-frame form. We show that with a Noether symmetry the
corresponding dynamical system can be completely integrated and the potential exhibited by the symmetry
can be exactly obtained. With the proper choice of parameters, the behavior of the scale factor displays an
exponential (de Sitter) behavior at the present epoch. Moreover, we discover that the Hubble parameters
strongly depends on the initial values of parameters exhibited by the Noether symmetry. Interestingly, it
can retardedly evolve and becomes a constant in the present epoch in all cases.
© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
*
Corresponding author.
E-mail
addresses: naragorn .k @psu .ac .th (N. Kaewkhao), thanyagamon1995 @gmail .com (T. Kanesom),
channuie @gmail .com (P. Channuie).
https://doi.org/10.1016/j.nuclphysb.2018.04.011
0550-3213/© 2018
The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
N. Kaewkhao et al. / Nuclear Physics B 931 (2018) 216–225 217
1. Introduction
In modern cosmology, the mechanism of cosmic inflation seems conceivable. Inflation marks
nowadays an inevitable ingredient when studying very early evolution of the universe. The reason
stems from the fact that it solves most of the puzzles that plague the standard Big Bang theory,
and simultaneously is consistent with recent observational data. In other wo
rds, it not only gives
sensible explanations for the horizon, flatness, and relic abundant problems, but also provides
us primordial density perturbation as seeds of the formation for a large-scale structure in the
universe.
Despite all its success, ho
wever, the underlying mechanism of the inflationary physics is still
unknown. Recent observational data much flavors large field inflationary models with plateau-
like inflaton potentials. However, there are several different ways of constructing successful
inflationary models. In the supergravity context, a model with plateau potentials was proposed
by the authors of Refs.
[1,2]. Here it describes a potential which is exponentially approaching a
positive constant for super-Planckian values of the inflaton field. Later on, a theory with a similar
potential was realized as the Starobinsky model [3], and then the Higgs inflation model with a
similar potential wa
s developed [4,5]. It is worth noting that these models lead to nearly identical
predictions, providing the best fit to the latest Planck data [6,7]. Moreover, recent investigation
shows the inflaton field can emerge as a composite state of a new strongly interacting gauge
theory [8,9]. More recently
, a broad class of inflationary models, dubbed cosmological attractors
[10–16], yields very similar inflationary predictions.
Interestingly, the cosmological α-attractors incorporate most of the e
xisting inflationary mod-
els with plateau-like potentials including the Starobinsky model and some generalized versions
of the Higgs inflation. Regarding the α-attractors, the flatness of the inflaton potential is imple-
mented and protected by the existence of a pole in the kinetic term of the scalar field. M
oreover,
at large-field values, any non-singular inflaton potential acquires a universal plateau-like form
when performing the (conformal) transformation. Regarding the hyperbolic geometry and the
flatness of the Kahler potential in the supergravity context, the universal behaviors of these
theories make very similar cosmological predictions preserving good match to the latest cos-
mological observ
ations [17–20]. The successful extensions of these models can also describe
dark energy/cosmological constant and supersymmetry breaking [21–29].
The purpose of the present study is to analyze
the dynamics of the field in the cosmological
α-attractors through the Noether symmetry technique. Here we concentrate on the single field
scenario. It is worth noting that this approach proved to be very useful not only to fix physi-
cally viable cosmological models with respect to the conserv
ed quantities, e.g., couplings and
potentials, but also to reduce dynamics and achieve exact solutions. Moreover, the existence of
Noether symmetries plays crucial roles when studying quantum cosmology [30].
In the present paper
, we consider the single-field model of cosmological α-attractors. The
structure of the paper is as follows: In Sec. 2, we introduce the model in the Jordan frame (JF)
and transform the original action into the equivalent Einstein frame (EF) by applying the con-
formal transformation. In Sec. 3, we adopt the Noether symmetry approach to study the related
dynamical systems obtained from a point-lik
e Lagrangian. Here we can determine the form of
the undefined potential of the action by imposing the Noether symmetry. We also compute the
general solutions with the helps of the new coordinate system. M
oreover, we compute the Hubble
parameter and display plot as a function of time for some specific cases. Finally, we conclude
our findings in the last section.
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